1. Scattering RT Calculations The phase function is defined as the angular cross section per particle, σ(θ′, φ′,θ,φ), normalized to the angular cross section integrated over a complete solid angle: “We come spinning out of nothingness, scattering ′ ′ stars like dust.” - Jalal ad-Din Rumi (Persian Poet, ′ ′ σ(θ , φ ,θ,φ) p(θ , φ ,θ,φ)= . 1207-1273) ′ ′ 4π dΩ σ((θ , φ ,θ,φ))/4π R or, assuming “disoriented” particles such that We’ve considered solutions to the radiative trans- ′ ′ p(θ , φ ,θ,φ)= p(cos Θ), fer equation for the obvious situation in which there is no source function. We’ve also solved the radiative we have: transfer for an isotropic source function, by making σ(cos Θ) the common approximation that the atmosphere can p(cos Θ) = . be subdivided into plane parallel levels that are thin 4π dΩ σ(cos Θ)/4π enough that the temperature and composition can be R Thus: assumed to be constant within the layer. In this case p(cos Θ) the source function is not only isotropic but also con- dΩ = 1 (1) Z4 4π stant within the layer. π or 1 Here we consider that case where the source func- p(cos Θ) tion is neither constant, nor isotropic. This case ap- dcosΘ = 1 (2) Z−1 2 plies to the scattering of sunlight off of planetary at- mospheres, particularly for Titan, which has hazy at- The probability of a scattering event in a region of mosphere, with particles that are highly forward scat- solid angle of dΩ around Θ is tering. To address the non-constant nature of the p(cos Θ) source function, we again subdivide the atmosphere dΩ. (3) into homogeneous layers. To address the anisotropic 4π nature of the source function we discretize the angu- lar integration of the source function. However, it is important to note that there are other ways of solv- ing the radiative transfer equation, that don’t involve any of these quantization tricks. Scattered Light Incident Light Θ 1.1. Phase Function To quantify the scattering of particulates we de- Fig. 1.— Definition of the scattering angle, Θ. fine the phase function, p(θ′, φ′,θ,φ), which depends on the direction of the incident light, characterized by θ′ and φ′, and the direction of the outgoing radiation A particle’s phase function depends on it’s size rel- of direction θ and φ. As such the scattering charac- ative to the wavelength of incident light, λ, which is teristics of one oddly shaped particle depends on four defined by the size parameter, x: variables, and can be quite complicated. However, in 2πr x = , (4) an atmosphere the suspended particles usually have λ no preferred orientation. In this case, there is no pre- ferred orientation, and the scattering phase function where r is the size or radius. In addition the phase depends only on the scattering angle, i.e. the angle function depends on the particle’s shape. Fig. 2 (Θ) between the direction of incident and scattered shows how spherical particles display more forward radiation (Fig. 1). Exceptions to this rule are snow scattering the larger the size parameter. If the size pa- flakes, which become oriented, and raindrops, which rameter is small enough then light is scattered equally become pie-shaped, as they fall. in the forward and backward direction. This can be seen from Rayleigh scattering (named after Lord 1 And the source function for scattering is: jν σs ′ p(Θ) ′ ′ Sν (θ, φ)= = dΩ Iν (θ , φ ), (5) κν σν Z4π 4π where σν is the total extinction coefficient, equal to the sum of the scattering and absorption cross sec- tions, σs and σa. The term σs/σν = σs/(σs + σa) = aν is called the single scattering albedo and, less than unity, is the probability of a scattering for each ex- tinction. The probability for absorption is then 1-aν. The radiative transfer equation for multiple scatter- ing and emission can be written as: dIν = −Iν + [1 − aν ]Bν (T ) Fig. 2.— Phase functions for different x. dτ aν ′ ′ ′ + dΩ p(Θ)Iν (θ , φ ). (6) 4π Z4π Rayleigh, who in 1871 worked out its properties), We integrate the incident intensity (primes) over all which is the scattering of particles for which x << 1. solid angles to determine the contribution of scattered It characterizes therefore the scattering of sunlight by intensity into the particular solid angle of interest. the molecules in an atmosphere. For many problems, the atmosphere can be treated as a collection of overlying parallel layers, which if 1.2. Source Function for Scattering subdivided finely enough, are of uniform composi- tion and temperature. For such a plane-parallel at- From the definition of intensity of light, Iν , we can mosphere it is common to designate τ as the verti- write the energy incident normally on an area dA in cal optical depth such that the actual path of light ′ ′ ′ the direction (θ , φ ) , within the solid angle dΩ in through a layer depends on the zenith angle, θ, at time dt and frequency interval dν centered at ν is: which photons enter the layer. The optical depth of ′ ′ ′ ′ this slant path is then: dEν = Iν (θ , φ ) dA dt dν dΩ The energy scattered is attenuated by σsds (where τs = τ/|µ|, (7) ds is the path length and σs is the scattering cross section). where |µ| = |cos θ|. Now p(θ′, φ′; θ, φ)/4π is essentially the probability density function, as evident from Eq. 1. Therefore the incoming intensity of light multiplied by this proba- 2. Particle Scattering bility density tells us what fraction of light, coming ′ ′ Generally gas and particles do not scatter isotrop- from the direction (θ , φ ), is scattered into the solid ically. The phase function, scattering efficiency, and angle dΩ in the direction (θ, φ). The total scattering single scattering albedo depend on the size of the scat- energy emerging in a direction (θ, φ) from a volume terers relative to the wavelength of light, the shape of element dV =ds dA from all the radiation from all in- the particles, and for large particles, the indices of cident solid angles can then be derived by integrating refraction. Scattering can change the polarization of over the incoming incident angles: light, which then also must be considered in a full de- ′ p(Θ) ′ ′ scription of scattering effect. We will begin by defin- dEν = σ(ν) dV dt dν dΩ dΩ Iν (θ , φ ). Z4π 4π ing a few terms and then examine a few particular cases of the single scattering of light. The emission coefficient for scattering is then: dEν ′ p(Θ) ′ ′ jν = = σs dΩ Iν (θ , φ ). dV dt dν dΩ Z4π 4π 2 3. Rayleigh Scattering Rayleigh scattering describes the scattering of sun- light by gas molecules in the atmosphere, and was originally formulated by Lord Rayleigh (1871) to ex- plain the color and polarization of the light from the sky. More generally, it describes the scattering of light by particles much smaller than the wavelength of the light, and smaller than the wavelength of the light divided by the magnitude, |n|, of the index of refrac- tion (nr − ini). In fact, the scattering properties of light depend on the size to wavelength ratio. For this reason we define the size parameter, x as: 2πr x = . (5) λ For Rayleigh scattering, x|n| << λ. When such a small particle is exposed to E-M waves, every part of it experiences simultaneously the E-M field. As a result the particle becomes polarized, such that the negative charges are displaced from the more massive (and therefore more fixed) positive nu- cleus. As a result of the charge separation the par- ticle acquires a dipole moment, p, which oscillates because of the externally applied electric field1. The particle then radiates as a result of the accelerating charges. An oscillating dipole produces an oscillat- ing electric field, Escat, (and therefore an outward propagating EM wave), which is proportional to the frequency of the oscillation squared. Since the inten- sity, i.e. the emitted areal power, is proportional to the square of the amplitude of the E field, the result- ing cross section to Rayleigh scattering depends on wavelength roughly as frequency to the forth power, −4 i.e. σRay ∝ λ , which paints most sunsets (Fig. 2). The Rayleigh scattering optical depth for an atmosphere of a particular composition can be ob- tained from the following website: http : //pds − atmospheres.nmsu.edu/educationandoutreach For a particle of isotropic polarizability the induced Fig. 3.— Rayleigh scattering of vertically polar- dipole moment of the particle is in the same direction ized (top) and horizontally (middle) polarized as the externally applied electric field. Similar to a light scatter polarized light in the different di- classical dipole, the resulting radiation field, i.e. the rections shown. Unpolarized light is made up electric field of the scattered light, has an amplitude, the sum of equal amounts of the orthogonally E, that is proportional to the projection of the dipole polarized components. (From Petty) moment in the direction of the observer. Thus the intensity of the scattered light, which is proportional to E2 is proportional to sin2(θ), where θ is the angle 1A dipole moment equals the amount of the charge times the effective displacement distance of the charges. 3 are not entirely of isotropic polarizability. Also scat- tering due to clouds and aerosols does not polarize light, and therefore diminishes the polarization of the sky. The phase function of Rayleigh scattering looks rather like a symmetric potato, and depends on the scattering angle as: 3 2 I (cos Θ) = (1 + cos (Θ)).